Ch 05 - I’m Up and Down, and Round and Round9th Ganita Manjari

Circle

The strict locus of all points in a 2-D plane that are exactly equidistant from a given fixed point (the centre).

Radius ($r$)

The constant, fixed distance from the centre to any point lying on the circle's boundary.

Chord

A straight line segment joining any two distinct points strictly on the circle.

Diameter ($d$)

A chord that passes perfectly through the centre, functioning as the absolute longest chord ($d = 2r$).

Boundary Conditions / Domain

• Strictly assumes a 2-dimensional Euclidean plane.
• The radius must strictly be $r > 0$. If $r = 0$, the locus collapses into a degenerate 'point circle'.

Rotational Symmetry

A circle possesses complete/infinite rotational symmetry. Rotating the 2D shape by absolutely ANY angle $\\theta$ about its centre leaves it looking exactly identical.

Reflection Symmetry

A circle possesses infinite lines of reflection symmetry. EVERY single straight diameter acts as a perfect mathematical mirror line, mapping the semi-circles onto each other perfectly.

Boundary Conditions / Domain

• Assumes an ideal mathematical circle in a perfectly flat 2D plane without any marked points on its circumference.

Minor Arc

The geometrically smaller portion of the boundary. It subtends an interior central angle $\\theta$ strictly satisfying 0^\\circ < \\theta < 180^\\circ.

Major Arc

The geometrically larger portion of the boundary. It subtends a central reflex angle $\\theta$ strictly satisfying 180^\\circ < \\theta < 360^\\circ.

Boundary Transition

If the arc's central angle hits exactly 180^\\circ, it mathematically ceases to be minor or major and perfectly forms a semicircle.